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REPLIES 



BY 



PROF. LAWRENCE S. BENSON, 

Author of BENSON'S GEOMETRY, 
TO 

PROF. E. T. QUIMBY, 

Dartmouth College, Hanover, N. H, 

PROF. WM. CHAUVENET, LL.D., 

Washington University, St. Louis, Mo , 

PROF. ROBERT D. ALLEN, 

Kentucky .Military Institute, FarnnJale, Ky., 



AND 



PROF. A- T. BLEDSOE, LL.D., 

Formerly University of Virginia, now Editor Southern Review, Baltimore, Md. 



NEAV YORK: 

James S Burnton. .49 Grind Street. 

1ST2. 



REPLIES 

BY 

PROF. LAWRENCE S'. "BENSON, 

Author of BENSON'S GEOMETRY, 
TO 

PROF. E. T. QUIMBY, 

Dartmouth College, Hanover, ft. II , 

PROF. WM. CHAUTENET, LL.D., 

ATashington University, St. Louis, Mo., 

PROF. ROBERT D. ALLEN, 

Kentucky Military Institute, Farmdale, Ky., 
AND 

PROF. A. T. BLEDSOE, LL.D., 

Formerly University of Virginia, now Editor Southern Review, Baltimore, Md. 



NEW YORK : ^ 

James S. Burnton, 149 Grand Street. 

1872. 



ADVERTISEMENT. 



Benson's Geometry contains the Elements of Euclid and Le- 
gendre, — simplified and arranged to exclude the illogical Reduclio ad 
Absurdum, — a full and explicit treatise on Plane and Spherical Trigo- 
nometry, also interesting and valuable exercises in Elementary 
Geometry and Trigonometry. 

The endorsements it has received from prominent mathematicians 
in all parts of the country, and the close scrutiny and critical exami- 
nations given to it by other prominent mathematicians, through 
which it has successfully passed, entitle Benson's Geometry to be 
esteemed as the most correct, the most progressive and the most 
complete Geometry that has ever been published. 

Benson's Geometry is adapted as a Text Book for schools and 
colleges ; it is elegantly bound in sheep, and has already passed 
through several editions. Price $2.00 per single copy. A liberal 
discount to schools and colleges. 



FOB SALE BY ALL BOOKSELLEBS. 






* 



CRITICAL EXAMINATIONS 



OF 



BENSON'S GEOMETRY. 



Before entering upon the subject matter of this treatise, I will 
make some preliminary remarks. About ten years ago, I published 
my " Scientific Disquisitions Concerning the Circle and Ellipse/' 
in which I advanced new views on the circle, and demonstrated that 
the circle is the arithmetical mean between the circumscribed and 
inscribed squares. I also offered a reward of one thousand dollars 
to any one who refuted the demonstration. The reward is still stand- 
ing, although there were many claimants for it. One, by the nomme de 
plume of " Dalton," controverted my conclusion, and, in consequence, 
a lively and interesting correspondence ensued in some prominent 
Southern journals, when, finally, two committees of expert mathema-, 
ticians were appointed to decide upon the issues involved. These 
committees were: Major P. F. Stevens, Sup't of the South Carolina 
Military Academy, Prof. Charles S. Venable, now of the University 
of Virginia, and William B. Carlisle, Esq., editor of the Charleston 
Courier, for myself: and Prof. Lewis R. Gibbes, of the Charleston 
College, Prof. James H. Carlisle, of Wofford College, and Rev. J. 
S. Kidney, of St. Thaddeus' Academy, Society Hill, South Carolina, 
for my respondent. 

No decision was ever arrived at by these committees, and in March, 
1864, I went to Europe to lay my views before prominent mathema- 
ticians and the various scientific and philosophical societies there; 
but I did not obtain any decision from them. 

Soon after the late civil war, I came to New York, and proposed to 
remodel the Elements of Euclid so as to substitute direct demonstra- 
tions for the Reductio ad Absurdum. This plan received the approval 
of Prof. G. B. Docharty, LL. D., of the College of the City of New 
York, Hon. S. S. Randall, City Superintendent of the Board of Edu- 
cation of New York, Prof. J. G. Fox, Principal of the Free Schools 
of Cooper Union, New York Association for the Advancement of 
Science and Art, Scientific American, New York Tribune, and 
many mathematicians and scientific journals in all parts of the 
country. 



In giving the direct demonstrations in all cases, I had to consider 
the Reductio ad Absurdum of Euclid, whereby he endeavored to sustain 
the proposition " that the area of the circle is the rectangle on the 
serai-circumference and radius." See Benson's Geometry, pp. 139, 
140, 141, 142, 143. Now, if this proposition be true, it can be directly 
demonstrated, which all true propositions can be; and the fact that 
it is impossible of being directly demonstrated, is a positive proof 
that it is not true. And, in giving the direct demonstration for the 
problem of the circle, I was unavoidably led to a different result from 
the false proposition of Euclid. 

I will give a sketch of my first demonstration, which is totally 
different in method from those I published in Benson's Geometry; 
but they all give the same conclusion. 

It is well known to all students of Geometry that Hippocrates, of 
Chios, was the first geometer who quadrated any portion of the circle ; 
he effected the quadrature of the lune — the portion of a circle, in the 
form of a crescent, contained by the semi-circumference of one circle 
and the arc of the quadrant of another. 

(See Diagram to the Twenty-fourth Proposition of the Fifth Book, 
Benson's Geometry.) A D C is the lune, equivalent to the triangle 
A C B ; hence, we have the segment of the quadrant, A C, unquad- 
rated ; and since a similar construction can be made in the other half 
of the circle, ADC B, we have the entire circle quadrated with the 
exception of the circular space in the middle composed by two seg- 
ments of quadrants ; consequently, to effect the quadrature of the 
circle, we have simply to quadrate the segment of a quadrant. 

In view of this, I considered the nature of the segment of a quadrant: 
I saw that its properties were the arc, chord and altitude. Knowing 
that geometrical quantities are abstract things, and that Geometry is 
entirely a hypothetical science, I reasoned from the hypothesis that the 
arc of the segment of a quadrant is geometrically equivalent in length to 
the chord and altitude of the segment : which makes the arc equivalent 
to the side of the mean square between the circumscribed and inscribed 
squares ; and the circle equivalent to three times square of radius. 

These are the outlines of my first demonstration. The dissenters 
to it, without reasoning upon the course of argument, objected at 
once to the conclusion. 

To disprove which, " Dalton " cited the case of an inscribed 
dodecagon being three times square of radius, and added that it is 
impossible for a part to be equal to a whole. 

My reply was, that in reasoning about the circle, we must bear in 
mind the difference between it and a polygon; that where one is 
formed by a curve, the other is formed by a certain number of straight 
lines; and that, consequently, the circle can not be treated as a 



polygon ; hence, conclusions drawn from the nature of the curve 
must be incongruous with conclusions drawn from the nature of the 
straight line. In proof of my conclusion I showed that it belonged 
to the class of truths derived from the nature of the circle, amongst 
which I embraced the truths discovered by Archimedes, relating to 
the cone, sphere and cylinder, their convex surfaces, and the quadra- 
ture of the lune ; with all of these my conclusion agrees (See Ben- 
son's Geometry, pp. 166, 216, 217,) — an irrefragable proof that my 
conclusion is true; for it is the chief characteristic of geometrical 
truths that they all agree withypne another, however they may be 
combined together, provided that the reasoning is correct. 

Now, "Dalton" and the other dissenters overlooked the fact of 
the above agreements, and they fell into the blunder that logicians 
call fallacia accidentis. Certainly, when a conclusion agrees with 
numerous truths, that evidence of its truth can not be impaired 
because that conclusion is incongruous with another conclusion .derived 
from a totally different source. 

Another dissenter (J. H. Schultze) had two tin cups constructed^ 
one, round, 3J inches diameter, and 3 inches deep ; the other, square, 
12 inches perimeter, and 3 inches deep — inside measurements; they 
held same quantity of water. 

My demonstration makes a square of 12.08-f- inches perimeter 
equivalent to circle with 3J inches diameter. By the method of 
Euclid, we have perimeter of polygon equivalent to circumference, 
3J X 3.1415926+ = 10.9955741-]-, showing that in the matter of the 
perimeter my method is more practical^ correct than the method of 
Euclid. Again : a square of 12 inches perimeter is equivalent to 9 
square inches. My demonstration gives 9.1875 square inches, exactly, 
for the square equal to the circle, whereas the method of Euclid gives 
9.6211-}- square inches for the same square ; hence, it is clearly seen 
that my demonstration is more practically correct than the method of 
Euclid ; and had the mechanical construction of the tin cups been 
more perfect, the difference between it and my demonstration would 
be imperceptible. When I published these figures, they were a 
quietus to Mr. Schultze ; so that, scientifically and practically, I 
sustained my demonstration. 

I will now give the criticisms which have been made to the demon- 
strations published in Benson's Geometry. Prof. E. T. Quimby, 
Dartmouth College, Hanover, New Hampshire, has written me about 
thirty letters; and although they are eminently controversal, I have 
failed to discover one ill-tempered sentence, and his whole correspond- 
ence bears evidence of a desire to investigate for truth's sake. He 
directed his attention first to the matter of the excess that I claim the 



method of APPROXIMATION gives for polygons inscribed in the para- 
bola. He writes: " I have been looking at your approximation com- 
putations of the area of the parabola in which you get an excess 
above the § rectangle. I have not made the numerical computations, 
because I presume you have been over them times enough to secure 
accuracy. If you have made no mistake of this kind, I think the 
difficulty must be in not carrying out your work to a sufficient number 
of decimals. Carry it out to 15 decimals, and see if your excess is 
not less." 

In reply, I showed by examples that the sum obtained by adding 
together a series of decimals is greater when we have a greater 
number of decimal places ; and that, had I used fifteen decimals in 
ray computations, I would have obtained a greater excess than I 
obtained by using six decimal places only. 

The method of approximation is the only method geometers use 
to obtain the area of the circle, and this method gives 3.1415926, &c, 
square inches for the area of the circle, when the radius is unity. 

And since the area of the parabola has been geometrically deter- 
mined by Archimedes, therefore, when the method of approximation 
gives an excess above the area of the parabola for the area of inscribed. 
polygons, most evidently that method is unreliable for deter- 
mining THE AREAS OF THE CIRCLE AND OTHER CURVILINEAR SPACES. 

Prof. QuiMBYnext directed his attention to Corollary 2, Proposition 
17, Book 6, Benson's Geometry. He wrote : " In this corollary I see 
no flaw till you say, as an inference from the fact that the triangle 
BSN and the segment B N generate equivalent solids, ' consequently 
the segment B N and the triangle BSN are equivalent.' Now, this 
inference is not legitimate; and not only does it not follow that the 
segment and triangle are equivalent, but it does follow that they are 
not equivalent; for, since they generate equivalent solids, it must be 
true that the part of the segment not belonging to the triangle gene- 
rates a solid equivalent to that generated by the part of the triangle 
not belonging to the segment. Now, since these two parts, viz., seg- 
ment B T and area T S N, generate equivalent solids, the area of 
one, multiplied by the distance its center of gravity moves — that is, 
by the circumference of the circle described by its center of gravity — 
is equal to the area of the other multiplied by the circumference 
described by its center of gravity. But the center of gravity of T S N 
evidently describes a larger circumference than that of B T. Hence, 
the area of B T is greater than T S N. Hence, the area of the 
segment B N is more than the triangle B S N." 

In reply, I requested Prof. Quimby to slfow me by what principle 
of reasoning or method of demonstration he could sustain the propo- 
sition that the volume of a body of revolution is equivalent to the generating 



surface multiplied by the circumference of the circle described by the center of 
gravity of the generating surface. 

Prof. Quimby wrote: "Will you admit this proposition? The 
volume generated by any plane figure revolved about a line in its own 
plane is measured by the area of the figure revolved multiplied by 
the mean circumference described b} T its points. I mean by this, that 
the area of the revolved surface is to be multiplied by the average 
distance its points travel. If that distance is not the same distance 
its center of gravity moves, then it is the distance some other one of 
its points moves. I will state the proposition to correspond to this 
last idea : The volume generated by a plane figure revolved (as above) 
is measured by the area of the figure multiplied by some one point of 
the figure. Hence, to apply it to our revolved surfaces : Vol. 
B T = area B T X circ. of some point of B T; vol. TSN= area 
TSNX circ. of some point of TSN; . • . area B T X circ. of some 
point of B T = area TSNX circ. of some point of T S N. But 
circumference described by any point of B T must be less than that 
described by any 'point of T S N. .' . area B T > area T S N." 

My reply was that Prof. Quimby had entirely overlooked the point 
I urged, which was to know by what course of reasoning he could 
prove that a volume of revolution is the product of the generating 
surface and the circumference described by any point of the surface. 

Prof. Quimby, in lieu of demonstration, stated that it was univers- 
ally acknowledged by all mathematicians that "the solid generated 
by the revolution of a plane figure about an axis in its own plane is 
measured by the area of the figure into the circumference described 
by its center of gravity ." 

Surely, then, if this proposition be so universally acknowledged, its 
proof should be possible ; and I thought it strange that Prof. Quimby 
never gave it, when I asked him for it ; and I was ready to admit all 
his reasoning, when he proved it to me. If Prof. Quimby had no way 
to prove it, he had no right to use it, and his reasoning is untenable. 

I am perfectly aware that mathematicians universally acknowledge 
that proposition ; at the same time, I am aware that their argument 
for it is based upon the Reductio ad Absurdum reasoning ; and if the 
proposition be true, it can be demonstrated altogether directly, 
which can not be done: hence, prima facie, the proposition is false. 

Prof. Quimby took another tack. He asked my assent to the fol- 
lowing proposition, which I admitted : " When two equivalent surfaces 
are moved in a line perpendicular to their plane, or revolved, the one 
which moves farther generates the greater volume. " Prof. Quimby 
then wrote : " Let us look at your proof again. It is based on this, 
in your own words (Benson's Geometry, p. 165,): 'When we have 
equivalent solids generated upon the same radius, the generating sur- 



8 

faces are equivalent.' I have said this is not true ; but I will retract 
that statement so far as to say that its truth or falsity will depend 
upon the meaning you attach to the words 'upon the same radius/ 
With the meaning you evidently give those words, the statement is 
not true. That is to say, in your diagram in p. 164, the segment B T 
and the area TSN are not ' upon the same radius ' in any sense that 
can make the statement true; and yet your demonstration requires 
you to say that those areas are equivalent because they generate 
equivalent solids. Is it not so ? Now I claim : 

"1. You must make segment B T equivalent to T S N, to prove 
area of circle equal to 3 R 2 . 

" 2. You have shown B T and T S N to generate equivalent solids. 

" 3. Since every point of T S N is farther from the axis of revolution 
than any point of B T, T S N must be less than B T. 

"4. Hence, area of circle is not equivalent to three times square on 
the radius, but is more than 3 R 2 . 

" I am unable to conceive what you mean by saying that B T and 
TSN are on the same radius, and I am equally unable to conceive 
how yen can dispute my statements above and be sane on mathe- 
matical questions." 

The difficulty here with Prof. Quimby was about what I meant by 
the words "upon the same radius." I wrote him that I meant the 
radius of computation (See Benson's Geometry, p. 217.) I meant the 
radius by which the volumes generated by B T and T S N are com- 
puted. I asked him to compute those volumes without using the 
same radius. He could not do so, hence the volumes have the same 
radius of computation; hence, however unequally their positions 
may be from the axis of revolution, B T and T S N are correspond- 
ing PARTS OF VOLUMES GENERATED BY THE SAME CONDITIONS, that is, in 

the same plane, around the same axis, with the same radius: having 
the same base, the same altitude, and the same solidities. We obtain 
the contents of the volume generated by the segment B T: first, by 
finding the contents of the cylinder generated by the rectangle 
PNEB; secondly, taking two-thirds of these contents, we have the 
volume of the hemisphere, of which the volume generated by B T is 
a part ; thirdly, we find the contents of the frustum of the hemisphere 
made by a plane through the point T parallel to the base P N; 
fourthly, we find the contents of the cone generated by the triangle 
formed by the above plane, the line T B, and the axis of revolution : 
and lastly, we have the contents of the volume generated by B T, by 
taking the sum of the frustum and cone from the hemisphere. We 
obtain the contents of the volume generated by T S N, by taking the 
sum of the same frustum and cone from the volume generated b 
P N S B, which solid is computed by means of the radius of the base 



9 

of the cylinder generated by P N E B : hence, in all these computa- 
tions, we have the same radius as the basis of TnE calculations. 
Therefore, the volumes generated by B T and T S N are computed 
under the same conditions ; and since their computations are dependent 
upon their generation, they are, therefore, generated under the same 
conditions. Which must necessarily be, because these volumes are 
corresponding parts of the volumes generated by P N B and PNSB, 
which are generated by the same conditions. Consequently, the 

EQUIVALENCE BETWEEN P N B AND P N S B IS NOT IMPAIRED BY THE 
POSITION OF B T AND T S N. 

Prof. Quimby, after this, undertook to defend the Reductio ad 
Absurdum. He reasoned thus : " You reject the Reductio ad Absurdum, 
saying that it does not amount to proof. I think it does. I say 
nothing now of any particular theorem, but of the method in general. 
Suppose I do this. There are two magnitudes, A and B ; I show, by 
strict mathematical demonstration, that the assumption that A is 
greater than B leads to absurdity ; I show, in the same manner, that 
to assume A less than B leads to a similar absurdity: Have I not 
shown, then, that A can not be either greater or less than B, and 
therefore must be equal to B ? If to prove that A can not be any- 
thing else than B, is not proving that A is B, I don't know anything 
about demonstration. I freely admit that the direct demonstration of 
a theorem is usually preferable ; but I do not admit that the indirect 
is not logical. So much for the Reductio ad Absurdum." 

Had Prof. Quimby, without supposing, but " by strict mathematical 
demonstration" shown, that A can not be either greater or less than 
B, most undoubtedly he would have proven A equal to B. He would, 
in thai case, have given a very direct demonstration,' and would have 
himself repudiated the Reductio ad Absurdum. 

Now, taking the method of the Reductio ad Absurdum "in general:" 
It is reasoning from a supposition which we know to be false ; hence, 
as its name implies, it leads to absurdity, which can not be otherwise, 
since it is undoubtedly absurd throughout, from beginning to end. 

Now, I admit that when we have two magnitudes, A and B, A is 
either greater than, less than, or equal to B; and if A be shown to 
be neither greater nor less than B, A is then equal to B. 

In the Reductio ad Absurdum, it is supposed, in the first place, that 
B is greater than A, and, for " the sake of argument," (?) B is assumed 
to be equal to a third magnitude, C, which is known to be greater 
than B; hence, the conclusion is absurd ; and the inference is that 
because A can not be equal to C, A can not be equal to any magni- 
tude that exceeds B ; but no proof is given why A can not be equal 
to a magnitude less than C and greater than B, for it is not shown that 

exceeds B by less than any assignable magnitude. 



10 

In the second place, in the Reductio ad Absurdum, it is supposed that 
B is less than A, and, for " the sake of argument," (?) B is assumed to 
be equal to a fourth magnitude, D, which is known to be less than B ; 
hence, the conclusion is again absurd, and the inference is that because 
A can not be equal to 1), A can not be equal to any magnitude less 
than B; but no proof is given why A cannot be equal to a magnitude 
greater than D and less than B, for it is not shown that B exceeds D 
by less than any assignable magnitude. 

Because A and B are both less than C, and are both greater than 
D, it does not necessarily follow that A and B are equal, for it is pos 
sible for A and B to be both less than C, and to be both greater than 
D, and still be unequal to one another. Let C be 9, and D be 6, A 
can be 8, and B can be 7. Now, A and B are both less than C, and 
are both greater than D, still there is no equality between A and B. So 
that the Reductio ad Absurdum is not only inconclusive in demonstra- 
tion, but it is also illogical in its modus operandi. 

The sophistry of this reasoning is in supposing that A can not be 
either greater or less than B, because A and B are both less than C 
and are both greater than D; but no proof is given to show that C 
exceeds B by some assignable magnitude equal to the excess of C over 
A, and no proof that B exceeds D by some assignable magnitude 
equal to the excess of A over D. It is surprising that the insufficiency 
of the Reductio ad Absurdum should have been overlooked, although it 
is on record that some of the ablest mathematicians have objected to 
the Reductio ad Absurdum. 

In regard to the proposition that A is either greater than, less than, 
or equal to B, it will be more in accordance with " strict mathe- 
matical demonstration " to prove directly that A is equal to B, than 
to use false premises, circuitous processes and sophistical arguments 
which finally result in absurd conclusions. For, if A be equal to B, 
it can be proven so by direct demonstration ; and if A can not be 
proven by direct demonstration equal to B, it is evidently c.'ear that A 
and B are unequal. 

In Logic, there is the Negative Reasoning, which is perfectly valid, 
and it is a very useful instrument of investigation ; but it differs 
materially from the Reductio ad Absurdum, since it has no false premiss 
no circuitous process, no sophistical argument, and no absurd con- 
clusion. 

Prof. Quimbjt again wrote: "Only one thing more: I want to ask 
you why, in revolving this parabola and triangle, I do not prove, as 
you prove your area of the circle, that the parabola is |- of the 
rectangle of its ordinate and abscissa." [Here Prof. Quimbv drew a 
diagram of the semi-curve of a parabola, circumscribed by a 
rectangle.] " Let the curve A C be a portion of a parabola, of which 



11 

A D is the axis. Revolved about A D, it generates a paraboloid 
which is J of the cvlirder generated bv the rectangle B D. Take 
E C, \ of B C. Draw E A. The trapezoid AECD generates a solid 
3>- of the same cylinder, therefore the section of the parabola and the 
trapezoid are equivalent, because, 'about the same axis, and with the 
same radius of generation, they generate equivalent solids/ But the 
trapezoid is f of the rectangle; therefore the parabola is f of the 
rectangle." 

The functions of a cone, cylinder and sphere are the radius of com- 
putation and the altitude of the body. In the case of the paraboloid, 
the functions are the parameter and abscissa of the parabola, of which 
the latter is the radius of computation ; whilst the functions of the 
circumscribing cylinder are represented by the ordinate and abscissa 
of the parobola, of which the former is the radius of computation ; 
showing, very evidently, that the paraboloid and cylinder are not 
generated under the same conditions. The circle being greater than any 
other isoperimetrical curvilinear surface, and the parabola not being a 
figure of revolution, it will require a less generating surface than the 
parabola to produce a volume of revolution equivalent to the para- 
boloid under the radius and altitude of the cylinder. 

The parabola not being generated by revolution, the principles of 
revolution are not applicable to it ; and the paraboloid is generated 
by a section of the parabola which does not preserve a constant 
equality of radii ; hence, the conditions are not the same with any 
volume generated by a constant equality of radii as the sphere, 
cylinder and cone. 

The functions of the paraboloid are derived from the properties of 
the parabola, in connection with the principle of revolution ; but the 
properties of the parabola are so distinct from the properties of the 
circle, that the properties of the paraboloid have dissimilar conditions 
from those of the circumscribing cylinder, because it has been shown 
that the functions of the paraboloid and circumscribing cylinder are 
entirely distinct. 

But, in the case of the volumes generated by the trapezoid B S N P, 
and the quadrant BNP, (Benson's Geometry, p. 164,) the conditions 
are the same; hence, the same conclusion can not be derived by 

REVOLVING SECTIONS OF THE PARABOLA AND CIRCLE. 

In Prof. Quimby's last letter, he wrote: "I would like to know 
how you get around the conclusion of the Differential Calculus, which 
gives the area of the circle the same as the old method by polygons. 
If the Calculus is wrong there, how do we know it is right on the 
parabola, &c? " 

Calculus treats problems in the same manner that they are treated 
in Elementary Geometry, which is the system of rectilinear truths. And 



12 

in order for the circle to be treated in Calculus, it had to be regarded 
as a great number of short, straight lines. Now, the straight line and 
curve differ essentially ; and the circle and polygon have no properties 
in common ; therefore, the treatment of the circle in Elementary 
Geometry and Calculus is fundamentally wrong. The fact that Cal- 
culus gives the same conclusion that Elementary Geometry does in 
the case of the circle, does not confirm the conclusion of Elementary 
Geometry ; because Calculus is based upon the principles of Element- 
ary Geometry, and the same errors that have crept into the rudiments 
have permeated through the trunk and branches of geometrical 
science. 

The parabola is not treated in Elementary Geometry; but in Ana- 
lytical Geometry, the properties of the parabola are confined to the 
condition that makes the constant equality of two varying straight lines; 
While the algebraic, transcendal and exponential equations of the 
parabola are derived from the system of rectangular co-ordinates ; and in 
no case are the properties of the parabola derived by treating it? 
curve as a great number of short, straight lines ; hence, there is no con- 
flict with any definition, axiom, demonstration or principle of Geome- 
try ; consequently, the conclusions in regard to the parabola are 
legitimate. 

Prof. William Chauvenet, LL. D., Washington University, St. 
Louis, lately deceased, wrote me: "I shall be willing to try to con- 
vince you of the fallacy of your conclusion in regard to the area of 
the circle, if you will answer, categorically and correctly, the follow- 
ing questions : 

" Is or is not the area of a regular dodecagon inscribed in a circle, 
equal in area to three times the square on the radius of the circle ? 

" Is or is not that dodecagon less than the circle ? 

"These are questions which can be answered without recourse to 
the use of revolving surfaces, &c, and all the world can jud^e of the 
correctness of the answers." 

It will be noticed that the questions of Prof. Chauvenet embody 
the objections made by " Dalton " to the conclusion of my first 
demonstration referred to in the beginning of this treatise ; and these 
objections, and my answers to them, were submitted to the com- 
mittees to whom I have also referred. My reply to Prof. Chauvenet 
was the same in effect as my reply to " Dalton." 

In geometrical demonstration, the straight line and curve are used : 
and although there are many varieties of the curve, there is but one 
kind of straight line ; because, according to the definition of the 
straight line, there can be but one kind having the length always in the 
same direction: whilst curves do not have the length always in the same 



13 

direction ; consequently curves admit of innumerable changes. Hence, 
it is possible to give a definition embracing all straight lines, but 
impossible to give the innumerable individual characteristics of curves 
under one definition ; and since these characteristics of curves are so 
distinct, a separate definition for each is required before curves can be 
successfully treated in the Elements of Geometry. In the absence of 
these individual definitions for curves, geometers have been forced to 
confine themselves to the properties of the straight line, and construct 
the science of Geometry upon those properties. 

It is well known that parallel straight lines are a most powerful 
instrument in geometrical investigation ; and they bear prima facie 
evidence that curvilinear magnitudes will not admit of their applica- 
tion : for it is a well known fact that approximate results only are 
obtained when the reasoning from parallel straight lines isapplied to 
curvilinear magnitudes. Now, since the straight line is so different 
from the curve, conclusions derived from the properties of the former 
must necessarily be different from the conclusions derived from the 
properties of the latter; and the conclusions in the one case will 
appear antagonistic to the conclusions in the other case ; but these 
conclusions, drawn from sources so opposite, will be perfectly con- 
sistent, logical and valid in their own peculiar bearings. Hence, a 
logical conclusion obtained from certain conditions must not be con- 
founded with another logical conclusion obtained from diferent 
conditions ; because, when we reason from the peculiar nature and 
formation of the circle, we show for the circle certain properties 
which will be necessarily incongruous with the properties of rectilinear 
magnitudes ; and when we reason from parallel straight lines, we will 
show for rectilinear surfaces certain properties which will approximate 
only to curvilinear surfaces. Therefore, according to the principle of 
parallel straight lines, three times square of radius will represent the 
area of a regular inscribed dodecagon ; but, according to the nature 
and properties of the circle, three times square of radius will rep- 
resent the area of the circle. As paradoxical as it may appear, there 
is no conflict with the well known axiom that a pari is less than a tohole ; 
because these conclusions are derived independently of one another, 
and they relate to separate and distinct conditions. The fact that the 
conclusion, that the circle is three times square of radius, agrees with 
numerous established truths relating to the cone, sphere and cylinder 
and the lune must outweigh its disagreement with one truth relating 
to polygons, when we take in consideration that there is a peculiar 
connection between the circle and the cone, sphere, cylinder and lune, 
and there is no peculiar connection between the circle and polygon. 
Prof. Chauvenet remained silent after my reply. 



14 

Several letters passed between Prof. Robert D. Allen, Kentucky 
Military Institute, Farm dale, Ky., and myself, preparatory to a series 
of discussions upon the various geometrical subjects embraced by my 
views, with the intention of having them put into pamphlet form for 
general circulation. But, for some reason or another, nothing further 
was done, although I was always ready to do my part. I respectfully 
invite Prof. Allen's attention to the points I make against the 
Heductio ad Absurdum, given in my reply to Prof. Quimby's defence of 
it, and I shall be glad to hear from Prof. Allen on the subject. 

When I arrived in London, May, 1864, Prof. A. T. Bledsoe, LL. D.. 
who was formerly Professor of Mathematics in the University of 
Virginia, sent me an invitation to call on him, which I did ; and in 
course of conversation Prof. Bledsoe informed me that he had heard 
that I was about to publish a work on Geometry. I remarked to him 
that I have such a work in preparation. He then said, " If you will 
bring me the manuscript, I will examine it; and I will give you some 
advice about it ; and I will be very frank and candid with you." I 
thanked him for his kind offer, and told him that I would bring him 
the manuscript on the next day. The following day I handed Prof. 
Bledsoe the manuscript ; and after two or three days had elapsed, I 
called on Prof. Bledsoe to learn what he had to say. He said: 
" Young man, I advise you not to publish this work." I inquired 
" Why?" He responded, " Because you are wrong." I asked him, 
"Where am I wrong?" He replied, "You are w T rong all over." I 
requested him to point out one place where I was wrong, and 
remarked : " I have a right to my opinion ; if you think me wrong, 
the burthen of proof rests with you ; and you should not denounce 
my work unless you can prove where I am wrong. I base my entire 
argument upon the single proposition that the arc of a quadrant is 
o-eometrically equivalent in length to the sum of the chord and altitude of 
the segment of the quadrant ; and if you disprove it, I will acknowledge 
that you have refuted my argument, and I will not publish anything 
more on the subject." He hemmed and hawed, and finally said : 
" You do not know anything about Geometry ; if you come to me, I 
will teach you Geometry in six months ; but you must give up all you 
now know and begin afresh." I thanked him for his "frankness " 
and "candor," and remarked: "So long as my proposition remains 
unrefuted, I will hold on to my own views." He rejoined: "You 
have nothing new : I taught the same ten years ago, at the University 
of Virginia." I replied: " If you taught the same years ago, it is 
very strange for you to make the objections you do." He requested 
me not to discuss the subject any more with him, remarking that the 
discussion would make him think of nothing else, and dream all 



15 ... . 

night about it: therefore, in our future interviews, this subject was 
eschewed from our conversation. 

A few years afterward, when I had published my Geometry, I 
received a letter from Prof. Bledsoe, who was then editor of the 
Southern Keview, in which he requested me to send him the names 
of those who approved of my Geometry, stating that he would make 
them "the subject of a long article in the pages of the Southern 
Review." I wrote him that I could not conceive in what way the 
names of the gentlemen who approved of my Geometry were con- 
cerned in the matter ; that when it suited me I would publish their 
names in my own way. I invited his attention to my arguments, and 
requested that he should make them "the subject of a long article in 
the pages of the Southern Review." 

A short time ago, I accidentally learned that Prof. Bledsoe pub- 
lished, in the winter of 1868-69, a long article in the Southern 
Review, severely criticising the gentlemen approving my views, my 
Geometry, and myself. 

I am surprised that Prof. Bledsoe, who was once Professor of 
Mathematics in the University of Virginia ; who is a doctor of laws ; 
who is the author of some books; who is the editor-in-chief of the 
Southern Review ; who was once a zealous laborer for the " Lost 
Cause;" and who filled some prominent positions under the Confed- 
erate Government, should publish such criticisms, circulate them, 
and not bring them to my notice. Such conduct appears to me very 
suspicious, and it resembles the assassin's trick — "a stab in the dark." 



ADDENDUM 



The attention of mathematicians is called to the following ; it is 
also to be found in Benson's Geometry, pp. 165, 166, 263, 264 : 

Circles are to one another as the squares described on their diam- 
eters (Benson's Geometry, V, 14) ; consequently, squares are to one 
another as the circles described on their sides (Benson's Geometry, 
V, 14, Cor. 2) ; hence, there is an equality of ratio between the circle 
and squares described about the same straight line ; hence, there is 
the same arithmetical relation between the circle and the inscribed 
square that there is between the circumscribed square and circle. To 
prove this, let the diameter be 10; circumscribed square is 100; 



16 

inscribed square is 50; and let the circle be x. The arithmetical 
progression is 50, x, 100 ; whence x — 50 = 100 — x. The geometrical 
proportion is 100 : x : : 50 : \x ; whence 100 — x = 2 (50 — %x). 

Substituting for 100 — x, its value from the arithmetical progres- 
sion, we have, x — 50 = 2 (50 — \x\ = 100 — x. That is, there is 
an agreement between the arithmetical progression and the geome- 
trical proportion. Now, the latter is unimpeachable; therefore, the 
former is also unimpeachable. 

The principles used here are the obvious ones, viz.: that the differ- 
ence between two consecutive terms of an arithmetical progression is equal to 
the difference between the next two consecutive terms ; and that the difference 
between the first and second terms of a geometrical proportion is as many 
times the difference between the third and fourth terms as is the ratio between 
the first and third terms. 

Hence, since x — 50 = 100 — x, we have x = 75. Or, the circle 

IS THE MEAN AREA BETWEEN TIIE CIRCUMSCRIBED AND INSCRIBED SQUARES. 

Hence, we have three distinct methods to prove the proposition 

THAT THE CIRCLE IS THE ARITHMETICAL MEAN BETWEEN THE CIRCUM- 
SCRIBED and inscribed squares, viz., by the Arc of the Quadrant, by 
the Eevolution of Surfaces, and by Proportion, — most convincing 
evidences of its truth, even independently of the fact that the propo- 
sition agrees with established truths of Geometry relating to the Cone,. 
Sphere, Cylinder and Lune. " Facts are stubborn things." 

Very respectfully, 

LAWRENCE S. BENSON. 
New York City, October, 1871. 



BENSON'S GEOMETRY 

EXCLUDES THE REDUCT1G AD ABSUIiDUM AND 
PROVES THE METHOD OF APPROXIMATION 
AN£KlU>& IN EXCESS. 



t is recommended by eminent Professors of mathematics, 
ieties of Science and is adopted by the Board of Education 
of New York and prominent Colleges and S bis. 



SEW YORK. 

BURNTON B ROTHERS. 

149 Grand Street. 



1/0 : TttUT H PREVAIL THOUGH THE HEAVENS SHOULD FALL. 



Cekiaik publishers alarmed by the success attending the introduction o' 
BENSON S GEOMETRY into the prominent College* and Schools, havp endeav- 
oured by a series of adverse articles in some educational journals to detract from 
the me -its of the work. They have siezed upon the difference between Cor 2. 
Prop. 17. Bk, 6. BENSON'S GEOMETRY, and the Reductio ad abturdum of 
Euclid and the Method of Approximation. 

The articles which appeared in the American Education al Monthly, are ba?ed 
upon the law found in Weisbaeh's Mechanics (Vol. 1, p 106.) in regard t< the 
revolution of surfaces around an axis. This law depends on the exact equality 
between the ratio, which we will call x, of the : ; reumference to the diameter, 
and the factor, which we will call y. which multiplied by the square of radius 
gives the area of the circle. The above. Monthly is unable to demonstrate the 
equality between x and y, but gives certain • processes ' by which they are show tt 
to differ, and adds "It is easy, by continuing the processes above, to prove .. 
nearer equality of x to y " The tenability of the position assumed by the Month- 
ly requires a positive proof of equality, and its subterfuge of '• a nearer equality" 
evinces the weakness of its position. Mathematics is the exact science niyj the 
utility of its deductions demands exactness to give correctness. For this reason, 
CARNOT in his Reflexions sur /a Me'aphysique du C afoul Infinitesimal 
states that those processes were not considered by the ancient geometers as consis- 
tent with the strictness of geometrical reasoning, and PLAYFAIR protested 
against those processes as follows. "In this way. also, the circumference and the 
area of the circle may be found still nearer the truth, but neither by this, nor 
by any other method yet known to geometers can they be exactly determined. 
although the errors of both may be reduced to a less quantity than any that can be 
assigned, " The method of reasoning pursued in BENSON'S GEOMETRY 
obviates approximate results and reduces the errors in the circumference and 
area of the circle to nothing. The American Journal of Mining takes exception to 
the same corollary, but its " argument " being a tirade of personal invectives and 
abuse, gives sufficient evidence of its own weakness. 

The Author of BENSON'S GEOMETKY has published a demonstration whereby 
the Method of Approximation, and the Reductio ad obsurdum reasoning of Euclid 
have been shown to make great errors in excess when applied to curvilineai spaces, 
consequently he has been in reeeipt of numerous communications from promi- 
nent mathematicians in various parts of the country and a f v extracts from the 
letters of Prof. E. T. Quimby. of Dartmouth College. Hanover, New Hampshire. 
will be pertinent on this occasion. In his letter dated Mav 25 th 1808. he states 
" I perfectly agree with you when you say that if the method nf appr ximation 
gives an area for the parabola greater than * the rectangb on the ordinate 
and abscissa, then either this method is wrong, or the demonstration of th tt 

1 



eorem ~'vi-g this area *c iV ~ - iral >la is fall ' ' ^.t the com >f this 

ter h<- writes: " I shal , enee nil we get the 

jht of the matter if it be your pleasure, and I desire that you would very freely 
>int out any errors I may make as I shal) wish to do any I may discover or 

nk I lis >ver in your work" Again, in his letter dated June 3rd 1868. 

wri es ! r iave been looking at your approximation computations of the area 
the paranoia in which you get an excess above the -§■ rectangle. I have not 
ide the numerical computation because I presume you have been over them 
lies enough to secure accuracy. If you have made no mistake ot this kind I 
ink the difficulty must be in not carrying out your work to a sufficient 
mber of decimals. Carry it out to 15 deci uals, and see if your excess is not less" 

reply, it was shown that where more decimals are used, by addition, the 
■ult becomes greater, and that if the work be carried out to 100 decimals, the 
cess would he greater than when 6 decimals only are used. In his letter dated 
me 24th 1868, he proposed to let the error in excess in case of approximation, 

" for the present :1 and turn his attention to the demonstration, proving the 
■cle | of the square of its diameter. In his letter dated July 16th 1868 he writes 
[ proposed to lay aside the question' of approximation for two reasons, 1, T had 
t time to make the calculations, and 2. I considered the other the main 
estion. " But in hi.s letter dated June 24th 1868. he gave the following propo- 
ion as an Axiom, '• When two equivalent surfaces are moved in a line perpen- 
iular to their plan:, or revolvjd, th.ic on; which moves farther generates the 

ater volume: " which is the same as Two Surfaces w r Hicn gkkep.itk equiva- 

NT SOLIDS WHBN T REVOLVED OX THE SAMS RADIUS, AND ABOUND THE SAME AXIS 

K themselves equivalext. Upon the latter proposition the demonstration 
the above corollary in BENSON'S GEOMETRY is based. 

But Prof. Quimby makes a particular application of his _4xiom and in 
ler to sustain his views, he asks assent to the following proposition in his letter 
ted July 8th 1868. viz " The volume generated by any plane figure revolved 
out a line in its own plane is measured by the area of the figure revolved, multi- 
ed by the mean circumference described by its points " This is in 
^stance the law quoted by the Am. Ed. Monthly from We isbach's Mechanics and 
likewise depends upon the equality of x. to y before mentioned, 
is incumbent upon Prof. Qui ruby ai-d the Amer. Edu Monthly, to prove the equal- 
ly x to y. which Prof Quimby admits in his letter dated July 16th 1868. in the 
lowing words — "I admit what you claim that it is my business to prove it so. 
vill write you again at length on the subject " But in his letter dated Oct 7 th 1868 
excuses himself from giviDg the proof on the pica that his time is so much occu- 
;d and the proof would be so long that it wc uld be an •' infliction ,: To prove t'ie 

2 



atWve equality; the Atner. Ed. MantTily "however, Tises the Method -or ifcppn 
nnation, which has been proven to giv* errors in excess, when applied to cove 
surfaces, The ancient geometer*, Playfair, Torelli, and other learned mod rn 
mathematicians hare protested against iLs use to determine the properties of the 

(circle, and Carkot in his Reflexion* stir ia Jictaphi/sique du Catcui lv 
mul after stating that the ancient geometers did not regard the Method of Approx i- 
anation as perfectly rigorous nor consistent with the strictness of geometrical 
reasoning, adds that, the continual approximation of these polygons 1<> tL ■ 
afforded an idea of the properties of the latter., but it still renaaius to be proved by 
some recognised principle of demonstration the truth of the properties that had 
thus in a manner been divined. For this reason A trc*iiMX4»ES proved tie' proper- 
ties of the parabola from the relation that certain rectangular figures ■described 
^ihout the parabola have to one another, and Torelli, who was so well \ -• ; in 
the reasoning of the great Greek geometer, proved by the same process of reason- 
ing that the eircle has to the square on its diameter exactly the ratio of •'• t > 1. 
PlaTFAIR who investigated Torelli's demonstration, admits that the pr-opositio.i 
upon which the demonstration is based, is true on .certain conditions. >\ hich 
.makes the demonstration perfectly valid, .sound and true, because the .fact <'t the 
propo-ition being true on.any conditions whatever shows that it contains tin' essen- 
tials of a geometrical proposition since every proposition of geometry is eondition- 
1'V true only; and a false proposition ian not .be true on any condition. Toredi's 
'demonstration is therefore perfectly legitimate, and being the same j> !•<.<•(_» of 
reasoning as that used by Archimedes lor the parabola, it is a recognised principle 
•of demonstration. As a collateral evidence tluit Torelli's demonstration is true, the 
Method of Revolution . another recognised principle of demonstration can "fee u>o(]. 
From Prof. Quimhy's axiom in his letter dated June 24 th 1868. l>efore given the 
conclusion derived by Torelli's demonstration is also obtained 

And Prof. Quimby in proposing to ''drop " the subject of-" th< «• in excess of 

approximation " and in evading the proof liis proposition in his letter dated July 8. 
1868. although he desired "to continue this correspondence "till we get the right 
-of the matter, " evidently found that the law quoted from Wctebach's Mechanic? 
is undemonstraole and untenable, nnd is consequently indefensible, null and void. 

BENSON'S GEOMETRY can b< btained from the principal Booksellers or by 
addressing BtTRNTON BROTHERS. 

149 Grand Street. NEW Y08K. 



MY VISIT TO THE SUN; 

Or Critical Essays on Physics, Metaphysics and Ethics, 

i 
By Prof. LAWRENCE S. BENSON, 

Author of BENSON'S GEOMETRY. 



Revised and Enlarged from the English Edition. 



IN PREPARATION FOR THE PRESS. 



Prof. G. B. Docharty, LL. D., of the College of the City of New 
York, writes : 

" I have had several interviews with Mr. Lawrence S. Benson on 
scientific subjects, and from his conversation, together with the essays 
which he has published, I esteem him an excellent ncholar and fine 
mathematician." 

" The views of the author are presented in the form of an allegory. 
He conceives himself on the Sun, where he surveys ihe universe; he 
pictures to himself the aspects, relations and conditions of things 
totally different from what they seem on the Earth ; and from his 
exalted standpoint he criticises the theories of Gravitation, Central 
Forces, Projectiles, Hypothetical Astronomy, Atomic Chemistry, 
Electricity, Galvanism and Magnetism ; he advances some new 
notions of Matter, and proclaims the seeming paradox that the Heat 
we feel in the sunlight is not emitted to us from the Sun, and that we 
never can obtain any knowledge of Cause. He promises, at some 
near future time, to dissipate the mists of Metaphysics and to solve 
some knotty questions in Ethics. The work is original in concep- 
tion, comprehensive in plan, bold in character, vigorous in style, 
exhaustive in research, forcible in argument, poetical in diction, and 
erudite in treatment. It challenges the scrutiny of savans ; it is 
readable and instructive, and will repay many perusals. We are 
anxious to see the continuance of it, as we regard it as a wonderful 
work, and we esteem it as a valuable acquisition to our stock of 
scientific lore." — Reviewer. 



Cji, 



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